Question 442319
Let's start off basic. Let's say you are dividing 4 into 43.

then 4 | 43     How many times does 4 go into 4?  1. So take 4*1 and subtract it from your first spot. Giving you:  How many times does 4 | 3?  0. So we multiply 4 by 0 and add the 3. Since we'd keep repeating the process we'd say the answer is 10, with a remainder of 3. This 3 can also be written as  3/4, the remainder over the divisor.

Let's apply this to polynomial division.

Divide  x+1 into 2x^2+4x-3.

How many times does x go into 2x^2?  {{{2x}}} times.

so 2x(x+1) = 2x^2 +2x and so we subtract that from our original.

So we are left with 2x -3.  We play the same game. How many times does x go into 2x?  {{{2}}} times. 

So we take 2(x+1)  = 2x+2 and we subtract that from our last total.  so [2x-3] - [2x+2] = -5.

We are left with a remainder of -5, but just like we did in our "numerical" example, we can write the remainder as a fraction. In this case, -5/x+1.

So our complete answer is {{{2x +2 - (5/(x+1))}}} for this made up problem. It's the same process, just they're xs.